Continuity and Differentiability

Continuity and Differentiability

This chapter explores the fundamental concepts of continuity and differentiability of functions, which are essential for understanding calculus. Students will learn how to determine if a function is continuous or differentiable at a point and over an interval, and how these properties are related.

1. Continuity

• A function f(x) is continuous at a point x = a if:
  • f(a) is defined
  • lim_x→a f(x) exists
  • lim_x→a f(x) = f(a)
• A function is continuous in an interval if it is continuous at every point in that interval.
• Types of discontinuities: removable, jump, and infinite.

Example:

2. Differentiability

• A function f(x) is differentiable at x = a if the derivative f'(a) exists.
• If a function is differentiable at a point, it is also continuous at that point (but not vice versa).
• Left-hand derivative and right-hand derivative must be equal for differentiability.

Example:

3. Important Results

• Every differentiable function is continuous, but every continuous function may not be differentiable.
• Algebra of continuous and differentiable functions (sum, difference, product, quotient).
• Chain rule for differentiation.

4. Practice Problems

1. Check the continuity of f(x) = 1/x at x = 0.
2. Is f(x) = x³ differentiable at x = 0?
3. Find the points of discontinuity of f(x) = [x] (greatest integer function).
4. Show that f(x) = sin(x) is continuous and differentiable everywhere.
5. If f(x) = x² + 3x + 2, find f'(x).

5. Summary

• Continuity: No breaks, jumps, or holes in the graph of the function.
• Differentiability: The function has a defined tangent (slope) at the point.
• Differentiability implies continuity, but not the other way around.